Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.946 + 0.323i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.766 + 0.642i)2-s + (0.597 + 0.802i)3-s + (0.173 − 0.984i)4-s + (−0.286 − 0.957i)5-s + (−0.973 − 0.230i)6-s + (−0.686 + 0.727i)7-s + (0.5 + 0.866i)8-s + (−0.286 + 0.957i)9-s + (0.835 + 0.549i)10-s + (0.835 − 0.549i)11-s + (0.893 − 0.448i)12-s + (−0.973 − 0.230i)13-s + (0.0581 − 0.998i)14-s + (0.597 − 0.802i)15-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + ⋯
L(s,χ)  = 1  + (−0.766 + 0.642i)2-s + (0.597 + 0.802i)3-s + (0.173 − 0.984i)4-s + (−0.286 − 0.957i)5-s + (−0.973 − 0.230i)6-s + (−0.686 + 0.727i)7-s + (0.5 + 0.866i)8-s + (−0.286 + 0.957i)9-s + (0.835 + 0.549i)10-s + (0.835 − 0.549i)11-s + (0.893 − 0.448i)12-s + (−0.973 − 0.230i)13-s + (0.0581 − 0.998i)14-s + (0.597 − 0.802i)15-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.946 + 0.323i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.946 + 0.323i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.946 + 0.323i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (12, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.946 + 0.323i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6059807753 + 0.1008172006i$
$L(\frac12,\chi)$  $\approx$  $0.6059807753 + 0.1008172006i$
$L(\chi,1)$  $\approx$  0.5882891383 + 0.2502154741i
$L(1,\chi)$  $\approx$  0.5882891383 + 0.2502154741i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.660240548358086333306491040438, −17.82999858332214717992914107503, −17.37400756578379416383066322899, −16.62946753244693529661099842668, −15.80009793984267968161610993304, −14.8259932303446063060087171369, −14.345057801480006777662235997952, −13.59369347645492505251446461410, −12.68879707980303195282630380485, −12.388066472798958569687734170273, −11.47064312776753216840700528711, −10.89168354744717906752724079349, −10.0326066107885677525481186193, −9.43611239513062551459205760220, −8.875116765186708309575181099580, −7.76033355706277263412328820216, −7.404353912851732788680697579717, −6.63037477352331262326531390499, −6.43724992852180386970062670381, −4.50585281571075900254132542594, −3.813364993854879016094180940410, −3.198392441252259306707973969432, −2.25714533921219925515913474337, −1.92838533232999921926802990792, −0.576551867746206372535891586554, 0.32608098224672048270144729764, 1.84047982178627585359598939464, 2.32772358955041797900019708208, 3.68083213477664773338814692881, 4.211848069652746858095224155133, 5.23151740724839333938157069198, 5.731684383227974303135672406094, 6.5588152563510684731261441548, 7.588990434504683082026736337, 8.27869317602089635205025895697, 8.83863464388799954130866943128, 9.3737880605797053316791542669, 9.77532726910071051039735134647, 10.77189932469314359220840457036, 11.43705225384190480318392389877, 12.37521408230325845896120770365, 13.10782827492475852400810280545, 13.96786765587041489324025015055, 14.70718255115368119749140594101, 15.44450117334012781363761468214, 15.62117288075216082761306343284, 16.6054064210135418637853424319, 16.890060048151474901896095153664, 17.51925517352982553988742675592, 18.68500945112018707468525596295

Graph of the $Z$-function along the critical line